In the present paper, we examine the existence of solutions to fourth-order differential equation systems when the L1-Carathéodory function is on the right-hand side. A concept of solution-tube for these issues is presented. The concepts of upper and lower solutions for fourth-order differential equations are extended to systems owing to this idea.
Citation: Bouharket Bendouma, Fatima Zohra Ladrani, Keltoum Bouhali, Ahmed Hammoudi, Loay Alkhalifa. Solution-tube and existence results for fourth-order differential equations system[J]. AIMS Mathematics, 2024, 9(11): 32831-32848. doi: 10.3934/math.20241571
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In the present paper, we examine the existence of solutions to fourth-order differential equation systems when the L1-Carathéodory function is on the right-hand side. A concept of solution-tube for these issues is presented. The concepts of upper and lower solutions for fourth-order differential equations are extended to systems owing to this idea.
In the present research, we prove existence results for a fourth-order differential equation system that takes the form:
{ϖ(4)(t)=f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t)),a.e. t∈J=[0,1],ϖ(0)=ϖ0,ϖ′(0)=ϖ1andϖ″∈(BC), | (1.1) |
where f:[0,1]×R4n→Rn represents an L1 -Carathéodory function, ϖ0,ϖ1∈Rn and (BC) can be the boundary conditions that are given by one of the following:
(SL) Strum-Liouville boundary conditions on J
A0ϖ(0)−β0ϖ′(0)=r0,A1ϖ(1)+β1ϖ′(1)=r1. | (1.2) |
(P) Periodic boundary conditions on J
ϖ(0)=ϖ(1),ϖ′(0)=ϖ′(1), | (1.3) |
where (Ai)i∈{0,1}∈Mn×n(R), such that
∀i∈{0,1},∃κi≥0:⟨ϖ,Aiϖ⟩≥κi‖ϖ‖2,∀ϖ∈Rn |
∀i∈{0,1},ri∈R:βi∈{0,1},κi+βi>0. |
We refer to [1,2,3] for further findings that were achieved in the specific instance of a boundary value issue for only one differential equation of the fourth-order (n=1), for more details, please see [4,5,6]. Existence results for higher-order differential equations can be found in [7,8], and the general case of Nth order systems is discussed in [9,10,11].
The concept of the solution-tube of problem (1.1) is presented in this work; see [12,13,14]. This idea is inspired by [15] and [16], where solution-tubes for second and third order differential equations systems are defined, respectively, as follows:
{ϖ″(t)=f(t,ϖ(t),ϖ′(t)),a.e. t∈J,ϖ∈(BC), | (1.4) |
and
{ϖ‴(t)=f(t,ϖ(t),ϖ′(t),ϖ″(t)),a.e. t∈J,ϖ(0)=ϖ0,ϖ′∈(BC). | (1.5) |
We prove that the system (1.1) has solutions. For this system, we employ the concept of a solution tube, which extends to systems the ideas of lower and upper solutions to the fourth-order differential equations presented in [17,18,19].
The structure of this paper is given as follows: This article will utilize the notations, definitions, and findings found in Section 2. In Section 3, we provide the idea of a solution-tube to get existence results for fourth-order differential equation systems. We then go on to demonstrate the practicality of our results through two examples.
In this section, we recall some notations, definitions, and results that we will use in this article. The scalar product and the Euclidian norm in Rn are denoted by ⟨,⟩ and ‖⋅‖, respectively. Also, let Ck(J,Rn) be the Banach space of the k-times continuously differentiable functions ϖ associated with the norm
‖ϖ‖k=max{‖ϖ‖0,‖ϖ′‖0,...,‖ϖ(k)‖0}, |
where
‖ϖ‖0=max{ϖ(t):t∈J}. |
The space of integral functions is denoted by L1(J,Rn), with the usual norm ‖⋅‖L1. The Sobolev space of functions in Ck−1(J,Rn), where k≥1 and the (k−1)th derivative is denoted by Wk,1(J,Rn).
For ϖ0,ϖ1∈Rn, we have the following:
Cϖ0(J,Rn):={ϖ∈C(J,Rn):ϖ(0)=ϖ0}, |
C1ϖ0,ϖ1(J,Rn):={ϖ∈C1(J,Rn):ϖ(0)=ϖ0,ϖ′(0)=ϖ1}, |
CkB(J,Rn)={ϖ∈Ck(J,Rn):ϖ∈(BC)}, |
Wk,1B(J,Rn))={ϖ∈Wk,1(J,Rn)):ϖ∈(BC)}, |
Ck+1ϖ0,B(J,Rn)={ϖ∈Ck+1(J,Rn):ϖ(0)=ϖ0,ϖ(k)∈(BC)}, |
Wk+1,1ϖ0,B(J,Rn))={ϖ∈Wk+1,1(J,Rn)):ϖ(0)=ϖ0,ϖ(k)∈(BC)}, |
Ck+2ϖ0,ϖ1,B(J,Rn)={ϖ∈Ck+2(J,Rn):ϖ(0)=ϖ0,ϖ′(0)=ϖ1,x(k)∈(BC)}, |
Wk+2,1ϖ0,ϖ1,B(J,Rn))={ϖ∈Wk+2,1(J,Rn)):ϖ(0)=ϖ0,ϖ′(0)=ϖ1,ϖ(k)∈(BC)}. |
Definition 2.1. A function f:J×R4n→Rn is called an L1-Carathéodory function if
(i) For every (ϖ,y,q,p)∈R4n, the function t↦f(t,ϖ,y,q,p) is measurable;
(ii) The function (ϖ,y,q,p)↦f(t,ϖ,y,q,p) is continuous for a.e. t∈J;
(iii) For every r>0, there exists a function hr∈L1(J,[0,∞)) such that ‖f(t,ϖ,y,q,p)‖≤hr(t) for a.e. t∈J and for all (ϖ,y,q,p)∈D, where
D={(ϖ,y,q,p)∈R4n:‖ϖ‖≤r, ‖y‖≤r, ‖q‖≤r, ‖p‖≤r}. |
Definition 2.2. A function F:C3(J,Rn)×J→L1(J,Rn) is integrally bounded, if for every bounded subset B⊂C3(J,Rn), there exists an integral function hB∈L1(J,[0,∞)) so that ‖F(ϖ,α)(t)‖≤hB(t), for ∀t∈J,(ϖ,α)∈B×J.
The operator NF:C3(J,Rn)×J→C0(J,Rn) will be associated with F and defined by
NF(ϖ)(t)=∫t0F(ϖ,α)(s)ds. |
We now state the following results:
Theorem 2.1. [20] Let F:C3(J,Rn)×J→L1(J,Rn) be continuous and integrally bounded, then NF is continuous and completely continuous.
Lemma 2.1. [21] Let E be a Banach space. Let v:J→E be an absolutely continuous function, then for
{t∈J:v(t)=0andv′(t)≠0}, |
the measure is zero.
Lemma 2.2. [22] For w∈W2,1(J;R) and ε≥0, assume that one of the next properties is satisfied:
(i) w″(t)−εw(t)≥0; for almost every t∈J,κ0w(0)−ν0w′(0)≤0,κ1w(1)+ν1w′(1)≤0; where κi,νi≥0,max{κi,νi}>0;i=0,1;andmax{κ0,κ1,ε}>0,
(ii) w″(t)−εw(t)≥0; for almost every t∈J,ε>0,w(0)=w(1), w′(1)−w′(0)≤0,
(iii) w″(t)−εw(t)≥0; for almost every t∈[0,t1]∪[t2,1],ε>0,w(0)=w(1), w′(1)−w′(0)≤0, w(t)≤0,t∈[t1,t2].
Then w(t)≤0, ∀t∈[0,1].
Lemma 2.3. [22] Let f∈C(J×R2n,Rn) be a L1-Carathéodory function (see definition in [22]). Consider the following problem:
{ϖ″(t)=f(t,ϖ(t),ϖ′(t)),a.e. t∈J,ϖ∈(BC). | (2.1) |
Let ε>0, and (z,N) a solution-tube of (2.1) given in Definition 2.3 of [22]. If ϖ∈W2,1B(J,Rn) satisfies
Π(t)=⟨ϖ(t)−z(t),ϖ″(t)−z″(t)⟩+‖ϖ′(t)−z′(t)‖2‖ϖ(t)−z(t)‖−⟨ϖ(t)−z(t),ϖ′(t)−z′(t)⟩2‖ϖ(t)−z(t)‖3−ε‖ϖ(t)−z(t)‖≥N″(t)−εN(t), |
a.e. on
{t∈J:‖ϖ(t)−z(t)‖>N(t)}. |
Then
‖ϖ(t)−z(t)‖≤N(t) for every t∈J. |
Now, we recall some properties of the Leray Schauder degree. The interested reader can see [23,24].
Theorem 2.2. Let E be a Banach space and U⊂E is an open bounded set. We define K∂U(¯U,E)={f:¯U→E, where f is compact and f(ϖ)≠ϖ, for every ϖ∈∂U}, the Leary-Schauder degree on U of (Id−f) is an integer deg(Id−f,U,0) satisfying the following properties:
(i) (Existence) If deg(Id−f,U,0)≠0, then ∃ϖ∈U, s.t.,
ϖ−f(ϖ)=0. |
(ii) (Normalization) If 0∈U, then deg(Id,U,0)=1.
(iii) (Homotopy invariance) If h:¯U×J→E is a compact such that ϖ−h(ϖ,α)≠0 for each (ϖ,α)∈∂U×J, then
deg(Id−h(.,α),U,0)=deg(Id−h(.,0),U,0), for every α∈J. |
(iv) (Excision) If V⊂U is open and ϖ−f(ϖ)≠0 for all ϖ∈¯U ∖V, then
deg(Id−f,U,0)=deg(Id−f,V,0). |
(v) (Additivity) If U1,U2⊂U are disjoint and open, such that ¯U=¯U1∪U2 and ϖ−f(ϖ)≠0 for all ϖ∈∂U1∪∂U2, then
deg(Id−f,U,0)=deg(Id−f,U1,0)+deg(Id−f,U2,0). |
In this section, we define the solution-tube to the problem (1.1). This definition is important for our discussion about the existence results. A solution to this problem is a function ϖ∈W4,1(J,Rn) satisfying (1.1). Now, we define the tube solution of problem (1.1), where the functions z∈W4,1(J,Rn) and N∈W4,1(J,[0,∞) are chosen before studying the existence of this problem.
Definition 3.1. Let (z,N)∈W4,1(J,Rn)×W4,1(J,[0,∞)). The couple (z,N) is solution-tube of (1.1), if
(i) N″(t)≥0,∀t∈J.
(ii) For almost every t∈J and for all (ϖ,y,q,p)∈F,
⟨q−z″(t),f(,t,ϖ,y,q)−z‴(t)⟩+‖p−z‴(t)‖2≥N″(t)N4(t)+(N‴(t))2, |
where
F={(ϖ,y,q,p)∈R4n:‖ϖ−z(t)‖≤N(t),‖y−z′(t)‖≤N′(t),‖q−z″(t)‖=N″(t),⟨q−z″(t),p−z‴(t)⟩=N″(t)N‴(t)}. |
(iii) z(4)(t)=f(t,ϖ,y,z″(t),z‴(t)),a.e.t∈[0,1] such that N″(t)=0 and (ϖ,y)∈R2n, such that ‖ϖ−z(t)‖≤N(t) and ‖y−z′(t)‖≤N′(t).
(iv) With (1.2), we have
‖r0−(A0z″(0)−β0z‴(0))‖≤κ0N″(0)−β0N‴(0), |
‖r1−(A1z″(1)+β1z‴(1))‖≤κ1N″(1)+β1N‴(1). |
If (BC) is given by (1.3), then
z″(0)=z″(1), N‴(0)=N″(1),‖z‴(1)−z‴(0)‖≤N‴(1)−N‴(0). |
(v) ‖ϖ0−z(0)‖≤N(0), ‖ϖ1−z′(0)‖≤N′(0).
The next notation will be used
T(z,N)={ϖ∈C2(J, Rn):‖ϖ″(t)−z″(t)‖≤N″(t),‖ϖ′(t)−z′(t)‖≤N′(t)and‖ϖ‴(t)−z‴(t)‖≤N″(t)forallt∈J}. |
The next hypotheses will be used:
(F1) f:J×R4n→Rn is a L1-Carathéodory function.
(H1) There exists (z,N)∈W4,1(J,Rn)×W4,1(J,[0,∞)) a solution-tube of the main system (1.1).
The next family of problems should be considered to prove the general existence theorem that will be presented:
{ϖ(4)(t)−εϖ″(t)=fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t)),a.e. t∈J,ϖ(0)=ϖ0, ϖ′(0)=ϖ1andϖ″∈(BC), | (3.1) |
where ε,α∈J and fεα:J×R4n→Rn is defined by
fεα(t,ϖ,y,q,p)={α(N″(t)‖q−z″(t)‖f1(t,ϖ,y,˜q,˘p)−ε˜q)−ε(1−α)z″(t)+(1−αN″(t)‖q−z″(t)‖)(z(4)(t)+N(4)(t)‖q−z″(t)‖(q−z″(t))),if‖q−z″(t)‖>N″(t),α(f1(t,ϖ,y,q,p)−εq)−ε(1−α)z″(t)+(1−α)(z(4)(t)+N(4)(t)N″(t)(q−z″(t))),otherwise, |
where (z,N) is the solution-tube of (1.1),
f1(t,ϖ,y,q,p)={f(t,ˉϖ,ˆy,q,p), if ‖ϖ−z(t)‖>N(t) and ‖y−z′(t)‖>N′(t),f(t,ϖ,y,q,p), otherwise, |
ˉϖ(t)=N(t)‖ϖ−z(t)‖(ϖ−z(t))+z(t), | (3.2) |
ˆy(t)=N′(t)‖y−z′(t)‖(y−z′(t))+z′(t), | (3.3) |
˜q(t)=N″(t)‖q−z″(t)‖(q−z″(t))+z″(t), | (3.4) |
˘p(t)=p+(N‴(t)−⟨q−z″(t),p−z‴(t)⟩‖q−z″(t)‖)(q−z″(t)‖q−z″(t)‖), | (3.5) |
and where we mean
N(4)(t)N″(t)(q−z″(t))=0on{t∈J:‖q(t)−z″(t)‖=N″(t)=0}. |
We associate with fεα the operator Fε:C3(J,Rn)×J→L1(J,Rn) defined by
Fε(ϖ,α)(t)=fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t)). |
Similarly to the Lemma 3.3 and Propositions 3.4 in [20] and results in [25], we need the following auxiliary results:
Lemma 3.1. Assume (H1). If a function ϖ∈W4,1ϖ0,ϖ1,B(J,Rn) satisfies
⟨ϖ″(t)−z″(4)(t)⟩+‖ϖ‴(t)−z‴(t)‖2‖ϖ″(t)−z′′(t)‖−⟨ϖ′′(t)−z′′(t),ϖ′′′(t)−z′′′(t)⟩2‖ϖ′′(t)−z′′(t)‖3−ε‖ϖ′′(t)−z′′(t)‖≥N(4)(t)−εN′′(t), |
for a.e. t∈{t∈J:‖ϖ″(t)−z″(t)‖>N″(t)}, then ϖ∈T(z,N).
Proof. By assumption
ϖ′∈W3,1ϖ1,B(J,Rn), ϖ″∈W2,1B(J,Rn), |
and thus, from applying Lemma 2.3 to ϖ″, we obtain
‖ϖ″(t)−z″(t)‖≤N″(t),∀t∈J. |
On
{t∈J:‖ϖ′(t)−z′(t)‖>N′(t), ‖ϖ′(t)−z′(t)‖′≤‖ϖ″(t)−z″(t)‖≤N″(t).} |
The function
t→‖ϖ′(t)−z′(t)‖−N′(t), |
is nonincreasing on J. Since
‖ϖ′0−z′(0)‖≤N′(0), |
we get
‖ϖ′(t)−z′(t)‖≤N′(t),∀t∈J, |
hence
‖ϖ(t)−z(t)‖′≤‖ϖ′(t)−z′(t)‖≤N′(t). |
The function
t→‖ϖ(t)−z(t)‖−N(t), |
is nonincreasing on J and since
‖ϖ(0)−z(0)‖≤N(0), |
we obtain
‖ϖ(t)−z(t)‖≤N(t),∀t∈J. |
Proposition 3.1. Assume (F1) and (H1) hold. Then the operator Fε that was defined earlier is continuous and integrally bounded.
Proof. First, we will prove that Fε is integrally bounded. If ϖ∈B, where B is a bounded set of C3(J,Rn), ∃K>0 that satisfies ‖ϖ(i)(t)‖≤K, ∀t∈J, where i=0,1,2,3. Then fεα(t,.,.,.,.) is bounded in E, it can be observed that
‖Fε(ϖ,α)(t)‖=‖fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))‖≤max{‖f(t,ϖ,y,q,p)‖,(ϖ,y,q,p)∈E}+|N″(t)|+‖z″(t)‖+‖z(4)(t)‖+|N(4)(t)|, |
for all α∈J and almost every t∈J, where
E={(u,y,q,p)∈R4n:‖u‖≤‖z‖0+‖N‖0,‖y‖≤‖z′‖0+‖N′‖0,‖q‖≤‖z″‖0+‖N″‖0,‖p‖≤2‖ϖ‴‖0+‖z‴‖0+‖N‴‖0}. |
As f is L1-Carathéodory, z∈W4,1(J,Rn) and N∈W4,1(J,[0,∞)), it is easy to see that Fε is integrally bounded.
In order prove the continuity, we should firstly prove that if (ϖp,αp)→(ϖ,α) in C3(J,Rn)×J, then
fεαp(t,ϖp(t),ϖ′p(t),ϖ″p(t),ϖ‴p(t))→fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))a.e. t∈J. | (3.6) |
Using the fact that f is L1-Carathéodory, and from the definition of fεα, it can be concluded that (3.6) is true a.e. on {t∈J:‖ϖ″(t)−z″(t)‖≠N″(t)}. Then, by Lemma 2.1 and Proposition 3.5 in [22], we easily show that ˘ϖ‴n(t)→ϖ‴(t) on
{t∈J:‖ϖ″(t)−z″(t)‖=N″(t)>0}, |
where ˘ϖ‴n(t), is defined as (3.5). Then, (3.6) is satisfied on
{t∈J:‖ϖ″(t)−z″(t)‖=N″(t)>0}. |
For
A={t∈J:‖ϖ″(t)−z″(t)‖=N″(t)=0}, |
where ϖ″(t)=z″(t), and by Lemma 2.1, it is not hard to see that ϖ‴(t)=z‴(t), N‴(t)=0 and N(4)(t)=0, ∀t∈A, which means,
fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))=α(f1(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))−εϖ″(t))+(1−α)(z(4)(t)−εz″(t))=αf1(t,ϖ(t),ϖ′(t),z″(t),z‴(t))−εz″(t)+(1−α)z(4)(t), |
a.e. on A. By the solution tube hypothesis (Definition 3.1 condition (iii)), we have
fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))=αz(4)(t)+(1−α)z(4)(t)−εz″(t)=z(4)(t)−εz″(t), |
a.e. on A. Consequently, (3.6) must be true a.e. on J. Using the Lebesgue-dominated convergence theorem, and since Fε is integrally bounded, the proof can be concluded.
Now, we can obtain our general existence result. We follow the method of proof given in [20].
Theorem 3.1. Assume (F1), (H1), and the following conditions are satisfied:
(Hk) For every solution ϖ of the related system (3.1), ∃K>0, so that
‖ϖ‴(t)‖<K,∀t∈J. |
Then, problem (1.1) has a solution ϖ∈W4,1(J,Rn)∩T(z,N).
Proof. We first show that if (ϖ,N)∈W4,1ϖ0,ϖ1,B(J,Rn)×W4,1(J,[0,∞)) is a solution of (3.1), then
‖ϖ″(t)−z″(t)‖≤N″(t),∀t∈J. |
For the set
{t∈J:‖ϖ″(t)−z″(t)‖>N″(t)}. |
By the definition of ~ϖ″ and ˘ϖ‴(t) (as (3.4) and (3.5)), we have
‖~ϖ″(t)−z″(t)‖=N″(t), | (3.7) |
<~ϖ″(t)−z″(t),˘ϖ‴(t)−z‴(t)>=N″(t)N‴(t). |
Also
‖˘ϖ‴(t)−z‴(t)‖2=‖ϖ‴(t)−z‴(t)‖2+(N‴(t))2−⟨ϖ″(t)−z″(t),ϖ‴(t)−z‴(t)⟩2‖ϖ″(t)−z″(t)‖2. |
Then, by (H1), we obtain
⟨ϖ″(t)−z″(t)−z(4)(t)⟩+‖ϖ‴(t)−z‴(t)‖2‖ϖ″(t)−z″(t)‖−⟨ϖ″(t)−z″(t),ϖ‴(t)−z‴(t)⟩2‖ϖ″(t)−z″(t)‖3−ε‖ϖ″(t)−z″(t)‖=⟨ϖ″(t)−z″(t),fεα(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))+εϖ″(t)⟩‖ϖ″(t)−z″(t)‖+1‖ϖ″(t)−z″(t)‖(‖ϖ‴(t)−z‴(t)‖2−⟨ϖ″(t)−z″(t),ϖ‴(t)−z‴(t)⟩2‖ϖ″(t)−z″(t)‖2)−ε‖ϖ″(t)−z″(t)‖=⟨ϖ″(t)−z″(t),αN″(t)‖ϖ″(t)−z″(t)‖(f1(t,ϖ(t),ϖ′(t),~ϖ″(t),˘ϖ‴(t))−z(4)(t))⟩‖ϖ″(t)−z″(t)‖+⟨ϖ″(t)−z″(t),(1−αN″(t)‖ϖ″(t)−z″(t)‖)N(4)(t)(ϖ″(t)−z″(t))‖ϖ″(t)−z″(t)‖⟩‖ϖ″(t)−z″(t)‖−ε⟨ϖ″(t)−z″(t),α(~ϖ″(t)−z″(t))−(ϖ″(t)−z″(t))⟩‖ϖ″(t)−z″(t)‖+‖˘ϖ‴(t)−z‴(t)‖2−(N‴(t))2‖ϖ″(t)−z″(t)‖−ε‖ϖ″(t)−z″(t)‖=α‖ϖ″(t)−z″(t)‖⟨~ϖ″(t)−z″(t),f1(t,ϖ(t),ϖ′(t),~ϖ″(t),˘ϖ‴(t))−z(4)(t)⟩+N(4)(t)(1−αN″(t)‖ϖ″(t)−z″(t)‖)−ε‖ϖ″(t)−z″(t)‖+ε‖ϖ″(t)−z″(t)‖−ε⟨ϖ″(t)−z″(t),α(~ϖ″(t)−z″(t))⟩‖ϖ″(t)−z″(t)‖+‖˘ϖ‴(t)−z‴(t)‖2−(N‴(t))2‖ϖ″(t)−z″(t)‖≥α‖ϖ″(t)−z″(t)‖(N″(t)+(N‴(t))2−‖˘ϖ‴(t)−z‴(t)‖2)+N(4)(t)−αN″(t)‖ϖ″(t)−z″(t)‖−αεN″(t)+‖˘ϖ‴(t)−z″(t)‖2−(N‴(t))2‖ϖ″(t)−z″(t)‖=N(4)(t)−αεN″(t)+(1−α)(‖˘ϖ‴(t)−z‴(t)‖2−(N‴(t))2)‖ϖ″(t)−z″(t)‖≥N(4)(t)−εN″(t), |
on
{t∈J:‖ϖ″(t)−z″(t)‖>N″(t)}. |
Using Lemma 3.1, it can be observed that any solutions to system (3.1) are in T(z,N) and then, in U, where
U={ϖ∈C3(J,Rn):‖u(i)‖0≤‖z(i)‖0+‖N(i)‖0+1,i=1,0,2;‖ϖ‴‖0≤K}. |
Fix ε∈J such that the operator Lε:C1B(J,Rn)→C0(J,Rn) given by
Lε(ϖ)(t)=ϖ′(t)−ϖ′(0)−ε∫t0ϖ(s)ds |
is invertible.
Consider the linear operator D:C3ϖ0,ϖ1,B(J,Rn)→C1B(J,Rn) defined by
D(ϖ)=ϖ″. |
It can be easily confirmed that D is invertible.
A solution to (1.1) is a fixed point of the operator
K=D−1oL−1εoNFε:C3(J,Rn)×J→C3ϖ0,ϖ1,B(J,Rn)⊂C3(J,Rn). |
Using Proposition 3.1 and Theorem 2.1, and since the operators D and Lε are continuous, it can be concluded that K is completely continuous and fixed point free on ∂U. Let
K0:C3(J,Rn)×J→C3(J, Rn) |
by K0(ϖ,α)=αK(ϖ,0). Because Fε(.,0) is integrally bounded, there exists an open bounded set K⊂C3(J,Rn), where
U⊂K and K0(C3(J,Rn)×J)⊂K, |
it can be implied from the homotopic and the excision properties of the Leray-Schauder theorem that
1=deg(Id,K,0)=°(Id−K0(.,1),K,0)=deg(Id−K(.,0),K,0)=°(Id−K(.,0),U,0)=deg(Id−K(.,1),U,0). |
As a result, there exists a solution ϖ∈T(z,N) for α=1 to (3.1), which also can solve (1.1) by definition of fε1. The proof is complete.
Now, following from our general existence theorem (Theorem 3.1), other existence results will be presented. We will consider the following assumptions:
(H2) There exist a function γ∈L1(J,[0,∞)) and a Borel measurable function Ψ∈C([0,∞),[1,∞)) s.t.
(ⅰ) ‖f(t,ϖ,y,q,p)‖≤γ(t)Ψ(‖p‖),∀t∈J and ∀(ϖ,y,q,p)∈R4n, where ‖ϖ−z(t)‖≤N(t), ‖y−z′(t)‖≤N′(t) and ‖q−z″(t)‖≤N″(t),
(ⅱ) ∀c≥0, we have
∫∞cdτΨ(τ)=∞. |
(H3) There exist, a function γ∈L1(J,[0,∞)) and a Borel measurable function Ψ∈C([0,∞],]0,∞)) s.t.
(ⅰ) ‖⟨p,f(t,ϖ,y,q,p)⟩‖≤Ψ(‖p‖)(γ(t)+‖p‖),∀t∈J and ∀(ϖ,y,q,p)∈R4n, where ‖ϖ−z(t)‖≤N(t), ‖y−z′(t)‖≤N′(t) and ‖q−z″(t)‖≤N″(t),
(ⅱ) ∀c≥0, we have
∫∞cτdτΨ(τ)+τ=∞. |
(H4) ∃r,b>0, c≥0 and a function h∈L1(J,R) s.t. ∀t∈J, ∀(ϖ,y,q,p)∈R4n, where
‖ϖ−z(t)‖≤N(t),‖y−z′(t)‖≤N′(t),‖q−z″(t)‖≤N″(t), |
and ‖p‖≥r, then
(b+c‖q‖)σ(t,ϖ,y,q,p)≥‖p‖−h(t), |
where
σ(t,ϖ,y,q,p)=⟨q,f(t,ϖ,y,q,p)⟩+‖p‖2‖p‖−⟨p,f(t,ϖ,y,q,p)⟩⟨q,p⟩‖p‖3. |
(H5) ∃a≥0 and l∈L1(J,R) s.t.
‖f(t,ϖ,y,q,p)‖≤a(⟨q,f(t,ϖ,y,q,p)⟩+‖p‖2)+l(t), |
∀t∈J and ∀(ϖ,y,q,p)∈R4n, where
‖ϖ−z(t)‖≤N(t),‖y−z′(t)‖≤N′(t), |
and
‖q−z″(t)‖≤N″(t). |
Theorem 3.2. Assume (F1), (H1), and (\mathcal{H}2) aresatisfied.If (BC) isgivenby (1.2) with \max \left\{ \beta _{0}, \beta _{1}\right\} > $$ 0 $, then system $ (1.1) $ has at least one solution $ \varpi \in T(z, N)\cap W^{4,1}( \mathcal{J}, \mathbb{\ R}^{n}) $.
Proof. Theorem 3.1 will guarantee the existence of a solution if we can obtain a priori bound on the third derivative of any solution ϖ to (3.1). It is known that ϖ∈T(z,N) from the Theorem 3.1 proof. Therefore, since (BC) is given by (1.2) with max{β0,β1}>0, ∃k>0, s.t.
min{‖ϖ‴(0)‖,‖ϖ‴(1)‖}≤k. |
Now, let R>k such that
∫RkdsΨ(s)>L=‖γ‖L1+ε‖N″‖0+‖z(4)‖L1+‖N(4)‖L1. |
Suppose there exists t1∈[0,1] s.t. ‖ϖ‴(t1)‖≥R. Then, there exists t0≠t1∈[0,1] such that ‖ϖ‴(t0)‖=k and ‖ϖ‴(t)‖≥k, ∀t∈[t0,t1]. Let us assume that t0<t1. Thus, by (H2), almost everywhere on [t0,t1], we have
‖ϖ‴(t)‖′=⟨ϖ‴(t)⟩‖ϖ‴(t)‖≤‖ϖ(4)(t)‖≤‖f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))‖+ε‖ϖ″(t)−z″(t)‖+‖z(4)(t)‖+|N(4)(t)|≤‖γ(t)‖Ψ(‖ϖ‴(t)‖)+ε‖N″(t)‖0+‖z(4)(t)‖+‖N(4)‖L1. |
So,
∫t1t0‖ϖ‴(t)‖′tΨ(‖ϖ‴(t)‖)dt≤L. |
Then, we have
∫t1t0‖ϖ‴(t)‖′tΨ(‖ϖ‴(t)‖)dt=∫‖ϖ‴(t1)‖‖ϖ‴(t0)‖dsΨ(s)≥∫RkdsΨ(s)>L, |
which contradict the assumptions. So, for any solution ϖ of (3.1), ∃R>0 s.t. ‖ϖ‴(t)‖<R, ∀t∈J.
If (H2) is replaced by (H3), extra assumptions are needed.
Theorem 3.3. Assume (F1), (H1), (H3), and (\mathcal{H}4) or (\mathcal{H}5) aresatisfied.Then,thereexistsasolution \varpi \in T(z, N)\cap W^{4,1}(\mathcal{J}, \mathbb{\ R}^{n}) to (1.1) $.
For this end, we need the next three Lemmas.
Lemma 3.2. [20] Let r,k≥0, N∈L1([0,1],R) and Ψ∈C([0,∞[,]0,∞[) be a Borel measurable function s.t.
∫∞rτdτΨ(τ)>‖N‖L1([0,1],R)+k. |
Then ∃K>0, s.t. ‖ϖ′‖0<K, ∀ϖ∈W2,1([0,1],Rn) satisfy:
(i) mint∈[0,1]‖ϖ′(t)‖≤r;
(ii) ‖ϖ′‖L1([t0,t1],R)≤k for every interval [t0,t1]⊂{t∈[0,1]:‖ϖ′(t)‖≥r};
(iii) |⟨ϖ′(t),ϖ″(t)⟩|≤Ψ(‖ϖ′(t)‖)(N(t)+‖ϖ′(t)‖) a.e. on
{t∈[0,1]:‖ϖ′(t)‖≥r}. |
Lemma 3.3. [20] Let r,ν>0, γ≥0 and N∈L1([0,1],R). Then there exists a nondecreasing function ω∈C[0,∞[,[0,∞[) s.t.
‖ϖ′‖L1([t0,t1],R)≤ω(‖ϖ‖0), |
and
mint∈[0,1]‖ϖ′(t)‖≤max{r,;ω(‖ϖ‖0)}. |
∀u∈W2,1([0,1],Rn) and
{t∈[t0,t1]:‖ϖ′(t)‖≥r}, |
the following inequality
(ν+γ‖ϖ(t)‖)σ0(t,ϖ)+γ⟨ϖ(t),ϖ′(t)⟩2‖ϖ(t)‖ϖ′(t)‖≥‖ϖ′(t)‖−N(t) |
is satisfied, where
σ0(t,ϖ)=⟨ϖ(t),ϖ″(t)⟩+‖ϖ′2‖ϖ′(t)‖−⟨ϖ′(t),ϖ″(t)⟩⟨ϖ(t),ϖ′(t)⟩‖ϖ′(t)‖3. |
Lemma 3.4. [20] Let K>0, and N∈L1([0,1],R). Then there exists an increasing function ω∈C([0,∞[,]0,∞[) s.t. ‖ϖ′‖L1([0,1],R)≤ω(‖ϖ‖0) for all ϖ∈W2,1([0,1],Rn) that satisfies
‖ϖ″(t)‖≤k(⟨ϖ(t),ϖ″(t)⟩+‖ϖ′(t)‖2)+N(t), |
for almost every t∈[0,1].
Proof of Theorem 3.3. Similarly to the previous proof, we need Theorem 3.1 to prove that the third derivative of all solutions ϖ to (3.1) is bounded. Let ϖ be a solution to (3.1), where ϖ∈T(z,N) from Theorem 3.1 proof. We obtain from (H3),
|⟨ϖ‴(t),ϖ(4)(t)⟩|≤|⟨ϖ‴(t),f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))⟩|+(ε‖ϖ″(t)−z″(t)‖+‖z(4)(t)‖+|N(4)(t)|)‖ϖ‴(t)‖≤(γ(t)+‖ϖ‴(t)‖)Ψ(‖ϖ‴(t)‖)+(ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)|)‖ϖ‴(t)‖≤(Ψ(‖ϖ‴(t)‖)+‖ϖ‴(t)‖)+(γ(t)+‖ϖ‴(t)‖+ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)|), |
for almost every t∈[0,1]. Thus, condition (iii) of Lemma 3.2 is verified, where
ψ(τ)=Ψ(τ)+τandN(τ)=γ(τ)+ε|N″(τ)|+‖z(4)(τ)‖+|N(4)(t)|. |
Therefore, it is enough to prove that conditions (i) and (ii) are verified. (H4) guarantees that a.e. on
{t∈[0,1]:‖ϖ‴(t)‖≥r}, |
we have
σ0(t,ϖ″)=⟨ϖ″(t)⟩+‖ϖ‴(t)‖2‖ϖ‴(t)‖−⟨ϖ‴(t)⟩⟨ϖ″(t),ϖ‴(t)⟩‖ϖ‴(t)‖3=ασ(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))+(1−α)‖ϖ‴(t)‖ +(1−α)⟨ϖ″(t)+(ε+N(4)(t)N″(t))(ϖ″(t)−z″(t))⟩‖ϖ‴(t)‖ −(1−α)⟨ϖ‴(t)+(ε+N(4)(t)N″(t))(ϖ″(t)−z″(t))⟩⟨ϖ″(t),ϖ‴(t)⟩‖ϖ‴(t)‖3≥ασ(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))+(1−α)‖ϖ‴(t)‖ −2(‖z″(t)‖+|N″(t)|)(‖z(4)(t)‖+ε|N″(t)|+|N(4)(t)|)r. |
Thus, we have
(b+c‖ϖ″(t)‖)σ0(t,ϖ″)+c⟨ϖ″(t),ϖ‴(t)⟩2‖ϖ″(t)‖‖ϖ‴(t)‖≥α‖ϖ‴(t)‖+b(1−α)‖ϖ‴(t)‖−h(t)−δ0,(t), |
where
δ0(t)=2r(b+c‖z″(t)‖+c|N″(t)|)(‖z″(t)‖+|N″(t)|)(‖z(4)(t)‖+ε|N″(t)|+|N(4)(t)|). |
If we take
z=minα∈[0,1]{α+b(1−α)},ν=bzandθ=cη, |
we can apply Lemma 3.3 to ϖ″([0,1],Rn). Thus, conditions of Lemma 3.2 are verified. Moreover, if (H5) holds, we have
‖ϖ(4)(t)‖≤α‖f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))‖+ε‖ϖ″(t)−z″(t)‖+‖z(4)(t)‖+|N(4)(t)|≤αa(⟨ϖ″(t),f(t,ϖ(t),ϖ′(t),ϖ″(t),ϖ‴(t))⟩+‖ϖ‴(t)‖2)+l(t) +ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)|≤a(⟨ϖ″(t),ϖ(4)(t)⟩+‖ϖ‴(t)‖2)+ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)| −a(1−α)⟨ϖ″(t),z(4)(t)+(N(4)(t)N″(t)+ε)(ϖ″(t)−z″(t))⟩≤a(⟨ϖ″(t),ϖ(4)(t)⟩+‖ϖ‴(t)‖2)+ε|N″(t)|+‖z(4)(t)‖+|N(4)(t)| +a(‖z″(t)‖+|N″(t)|)(‖z(4)(t)‖+|N(4)(t)|+εN″(t)). |
Therefore, if Lemma 3.4 is applied to ϖ″([0,1],Rn), all conditions of Lemma 3.2 are satisfied. As a result, for all solutions ϖ of (3.1), ‖ϖ‴‖0<K for some constant K>0.
From the previous results, we obtain the following consequence:
Corollary 3.1. Assume (F1), (H1), (H2), and (H4) or (H5) are satisfied. Then, we have a solution ϖ∈T(z,N)∩W4,1([0,1],Rn) to the system (1.1).
Remark 3.1. Definition 3.1 is associated to the definitions of lower and upper solutions to the fourth-order differential equation. These definitions are used in [17], and introduce them for problems (1.1) and (1.2).
Definition 3.2. Let n=1 and ϖ0=ϖ1=0.
A function κ∈C4(]0,1[)∩C3(J) is called a lower solution to (1.1) and (1.2), if
(i) κ(4)(t)≥f(t,κ(t),κ′(t),κ″(t),κ‴(t)) for every t∈J;
(ii) κ(0)=κ′(0)=0;
(iii) A0κ″(0)−β0κ‴(0)≤r0 and A1κ″(1)+β1κ‴(1)≤r1.
On the other hand, an upper solution to (1.1) and (1.2) is a function ν∈C4(]0,1[)∩C3(J) that satisfies (i)–(iii) with reversed inequalities.
Similarly to Remark 3.2 in [20], we consider the following assumptions:
(A) There exist lower and upper solutions, κ and ν, respectively, to (1.1) and (1.2), where κ≤ν .
(B) There exists a solution-tube (z,N) to (1.1) and (1.2).
(C) There exist lower and upper solutions, κ≤ν, to (1.1) and (1.2) s.t.
(i) κ″(t)≤ν″(t)) for all t∈J;
(ii) f(t,ν(t),ν′(t),q,p)≤f(t,ϖ,y,q,p)≤f(t,κ(t),κ′(t),q,p); ∀t∈J and (ϖ,y,q,p)∈R4n such that κ(t)≤ϖ≤ν(t) and κ′(t)≤ϖ′(t)≤ν′(t).
It can be easily checked that
● If (B) holds with z and N of class C4, and z(0)=N(0)=0, then (A) holds.
Indeed, κ=z−N and ν=z+N are respectively lower and upper solutions of (1.1) and (1.2). However, (A) does not imply (B).
Noting that (B) is more general than (C), see [17]; i.e.,
● If (C) is verified, then (B) is verified.
Taking z=(κ+ν)/2 and N=(ν−κ)/2. But, (B) does not imply (C) (ii) and κ(0)=ν(0)=0.
Next, we present two examples to illustrate the applicability of Theorem 3.3.
Example 3.1. Consider the following system:
{ϖ(4)(t)=ϖ‴(t)+‖ϖ‴(t)‖(‖ϖ″(t)‖2ϖ′(t)−⟨ϖ′(t),ϖ″(t)⟩ϖ″(t))−ξ,a.e. t∈J,ϖ(0)=0,ϖ′(0)=0,A0ϖ″(0)=0,A1ϖ″(1)+βt=1ϖ‴(1)=0, | (3.8) |
here ξ∈Rn,‖ξ‖=1, and Aiandβi are given before for i=0,1. Show that when z≡0,N(t)=t36, (z,N) is a solution-tube of (3.8). We have (H3) and (H4) are verified for
Ψ(τ)=3τ+1,γ(t)=0,b=1,c=0,r>0,h(τ)=2τr+τ5. |
Owing to the Theorem 3.3, the problem (3.8) has at least one solution ϖ s.t.
‖ϖ(t)‖≤t36, ‖ϖ(t)′‖≤t22 and ‖ϖ(t)′′‖≤t for all t∈J. |
Example 3.2. Consider the following system:
{ϖ(4)(t)=ϖ″(t)(‖ϖ‴(t)‖2+1)+φ(t),a.e. t∈J,ϖ(0)=0,ϖ′(0)=0,ϖ″(0)=ϖ″(1),ϖ‴(0)=ϖ‴(1), | (3.9) |
where φ∈L∞(J,Rn) with ‖φ‖L∞≤1. Show that for z≡0,N(t)=t22, (z,N) is a solution-tube of (3.9). We have (H3) and (H5) are verified when
Ψ(τ)=τ2+2, γ(t)=0,a=1,l(t)=3. |
By Theorem 3.3, the problem (3.9) has at least one solution ϖ s.t.
‖ϖ(t)‖≤t22, ‖ϖ′(t)‖≤t, ‖ϖ′′(t)‖≤1,∀t∈J. |
Our paper discusses the existence of solutions for fourth-order differential equation systems, focusing particularly on cases involving L1-Carathéodory functions on the right-hand side of the equations. We first, introduced the concept of a solution-tube, which is an innovative approach that extends the concepts of upper and lower solutions applicable to fourth-order equations into the domain of systems. It outlines the mathematical framework necessary to demonstrate that solutions exist for these types of differential equation systems under specified boundary conditions (such as Sturm-Liouville and periodic conditions). The paper stands on prior results regarding higher-order differential equations, providing a fresh perspective and methodology that can be used to explore further developments in the field. In addition to presenting the theoretical underpinnings, we also illustrated the practicality of our results with examples, contributing to the mathematical discourse on differential equations and our solutions, which ultimately serves as a scholarly contribution to understanding the dynamics of fourth-order systems and the existence of their solutions; please see [26,27].
Bouharket Bendouma: Conceptualization, formal analysis, Writing-original draft preparation; Fatima Zohra Ladrani and Keltoum Bouhali: investigation, Methodology; Ahmed Hammoudi and Loay Alkhalifa: Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.
The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).
The authors declare that there is no conflict of interest.
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